The divergence of E If the charge fills a volume, with charge per unit volume . R Where d is an element of volume. For a volume charge:

Thus: Gauss’s law in differential form.

Spherical polar coordinates (r, , ) r: the distance from the origin : the angle down from the z axis is called polar angle : angle around from the x axis is called the azimuthal angle

The Curl of E For a point charge situated at origin: Line integral of the field from some point a to some other point b: In spherical polar coordinates,

The integral around a closed path: Apply stokes theorem: True for electrostatic field.

Electric Potential Basic concept: The absence of closed lines is the property of vector field whose curl is zero. E is such a vector whose curl is zero. Using this special kind of it’s property we can reduce a vector problem: using V, we can get E very easily. Vector whose curl is zero, is equal to the gradient of some scalar function E=0  the line integral of E around any closed loop is zero (due to Stokes' theorem).

Therefore the line integral of E from point a to point b is the same for all paths. otherwise you could go out along path (i) and return along path (ii) and Because the line integral is independent of path, we can define a function O is some standard reference point. is called electric potential

The potential difference between two points a and b: Using fundamental theorem for gradients: So Electric field is the gradient of a scalar potential.

Electric Potential at an arbitrary point Electric potential at a point is given as the work done in moving the unit test charge (q0) from infinity (where potential is taken as zero) to that point. Electric potential at any point P is S.I. unit J/C defined as a volt (V) and 1 V/m = 1 N/C Note that Vp represents the potential difference dV between the point P and a point at infinity.

Potential Difference in Uniform E field Example: Uniform field along –y axis (E parallel to dl) When the electric field E is directed downward, point B is at a lower electric potential than point A. A positive test charge that moves from point A to point B loses electric potential energy. Electric field lines always point in the direction of decreasing electric potential.

Potential Diff. in Uniform E field Charged particle moves from A to B in uniform E field. q

Potential Diff. In Uniform E field (Path independence) Show that the potential difference between point A and B by moving through path (1) and (2) are the same as expected for a conservative force field. By path (1),

path (2) = 0 since E is  to dl

Equipotential Surfaces (Contours) VC = VB ( same potential) In fact, points along this line has the same potential. We have an equipotential line. No work is done in moving a test charge between any two points on an equipotential surface. The equipotential surfaces of a uniform electric field consist of a family of planes that are all perpendicular to the field.

Equipotential Surface Equipotential Surfaces (dashed blue lines) and electric field lines (orange lines) for (a) a uniform electric field produced by infinite sheet of charge, (b) a point charge, and (c) an electric dipole. In all cases, the equipotential surfaces are perpendicular to the electric field lines at every point.

Electrostatic Potential of a Point Charge at the Origin Q P 16

Electrostatic Potential Resulting from Multiple Point Charges Q2 P(R,q,f) Q1 O 17

Electrostatic Potential Resulting from Continuous Charge Distributions  line charge  surface charge  volume charge 18

Charge Dipole An electric charge dipole consists of a pair of equal and opposite point charges separated by a small distance (i.e., much smaller than the distance at which we observe the resulting field). d +Q -Q 19

Dipole Moment Dipole moment p is a measure of the strength of the dipole and has its direction. +Q -Q p is in the direction from the negative point charge to the positive point charge

Electrostatic Potential Due to Charge Dipole observation point d/2 +Q -Q z q P 21

P q d/2 d/2 22

first order approximation from geometry: q d/2 d/2 lines approximately parallel

Taylor series approximation: 24

In terms of the dipole moment: 25

Electric Potential Energy of a System of Point Charges q2 q1 and we know q3

The Energy of a Continuous Charge Distribution For a volume charge density p, Using Gauss’s Law: So: By doing integration by part: and so, If we take integral over all space:

Poisson’s and Laplace’s Equation The fundamental equations for E: Gauss’s law then says that: This is known as Poisson’s equation. In regions where there is no charge: Poisson’s equation reduces to Laplace’s equation. This is known as Laplace’s equation.